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let us consider Poisson process $N_t$ with $\lambda$ parameter and a stopping time $T=\inf\{t\ge 0;\,N_t=a\}$, where $a\in\mathbb{N}$. I would like to show that $ET=\frac{a}{\lambda}$, so I want to use Doob Theorem. I want to show that $P(T<\infty)=1$. What I have: $P(T=\infty)\le P(N_t<a,\,t\ge 0)\le P(N_t<a)=e^{-\lambda t}\sum_{k=0}^{a-1}\frac{(\lambda t)^k}{k!}\ldots$

I want to make some boundary for this sum independent of $t$ (don't know if it is even possible) and pass to the limit with $t\to\infty$. Any ideas?

mwr
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    Do you know about inter-arrival distribution? If you split arrival times of events, you will get $a$ exponential distribution with parameter $\lambda$ and $T$ is just sum of them. – Snowball May 28 '21 at 08:42
  • Of course! Hence it is $a/\lambda$ because it will be Gamma distribution. – mwr May 28 '21 at 09:33

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